Supreme Court nominee Brett M. Kavanaugh met with more senators Wednesday in his bid to win confirmation, while liberal opponents tried to link the nominee to a serial sexual harasser and even poked fun at his name.

Judge Kavanaugh met with Judiciary Committee members Orrin G. Hatch, Utah Republican, Lindsey Graham, South Carolina Republican, and other senators. Mr. Hatch, who called the nominee “a very fine man,” told reporters that he expects Judge Kavanaugh to pass the test.

“There will be the usual attempts to sully his reputation not only in the Senate but outside the Senate, but he’ll be able to handle it and I have every confidence he’ll be confirmed,” Mr. Hatch said.

With Democrats lacking the votes to stop the nomination outright, the efforts to go after the nominee’s reputation are well underway.

A women’s group called on senators to investigate Judge Kavanaugh’s knowledge of more than a dozen sexual harassment claims leveled at former 9th U.S. Circuit Court of Appeals Judge Alex Kozinski, for whom he worked as a law clerk in 1991.

“The American public deserves to know what Kavanaugh saw and heard, and if he did witness or hear about any harassment, what he did or could have done,” said Karin Roland, an official with the feminist group UltraViolet.

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Liberals also mocked the nominee’s name. “We’ll be DAMNED if we’re going to let five MEN — including some frat boy named Brett — strip us of our hard-won bodily autonomy and reproductive rights,” the pro-choice organization NARAL tweeted.

From Europe, where he was attending a NATO summit, President Trump jabbed at Democrats for being powerless in the confirmation battle. “If the Democrats want to win Supreme Court and other Court picks, don’t Obstruct and Resist, but rather do it the good ol’ fashioned way, WIN ELECTIONS!” the president tweeted.

Mr. Graham also chided Democrats for their tactics after meeting with the nominee. “If you’ve got a problem with President Trump’s [Supreme Court] picks, then you need to turn out and beat him next time,” Mr. Graham said.

Sen. Joe Manchin III, West Virginia Democrat, told a radio interviewer Wednesday that Judge Kavanaugh has “all the right qualities” to serve on the high court but said he hasn’t made up his mind how he will vote on the nomination.

Sen. Thom Tillis, North Carolina Republican and a member of the Judiciary Committee, predicted that a confirmation hearing would begin around the third week of August.

“There’s no reason why we shouldn’t do it,” Mr. Tillis told radio host Hugh Hewitt. “Get the documentation to everybody that wants it, let them read and be ready for the hearing. We have canceled [August] recess, and we’ve got time.”

Asked by Mr. Hewitt to rate on a percentage level his confidence that the Senate will confirm Judge Kavanaugh, Mr. Tillis replied, “100.”

As supporters and opponents search the nominee’s voluminous records from 12 years on the bench and service in the George W. Bush White House, The New York Times reported Wednesday that Deputy Attorney General Rod Rosenstein has asked federal prosecutors to help review the government documents of Judge Kavanaugh.

Mr. Rosenstein asked each of the 93 U.S. attorney’s offices to provide up to three federal prosecutors “who can make this important project a priority for the next several weeks.”